Analysis of biochemical reactions models with delays

Deterministic descriptions of three biochemical reaction channels formerly considered by Bratsun et al. (2005) [19] are studied. These descriptions are based on the mass action law and for the simple protein production with delayed degradation differ from that proposed by Bratsun et al. An explicit solution to this model is calculated. For the model of reaction with negative feedback and delayed production, global stability of a unique positive steady state is proved. According to the models of these two reaction channels considered in the present paper there cannot appear delayed induced oscillations. For the model of reaction with negative feedback, dimerisation and delayed protein production, local stability for a unique positive steady state is shown for some range of parameters. It is also proved that for some range of parameters the destabilisation due to the increasing delay can occur and delayed induced oscillations may appear.

[1]  M. Savageau.  Comparison of classical and autogenous systems of regulation in inducible operons , 1974, Nature.

[2]  L. Serrano,et al.  Engineering stability in gene networks by autoregulation , 2000, Nature.

[3]  C. Rao,et al.  Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm , 2003 .

[4]  D. Volfson,et al.  Delay-induced stochastic oscillations in gene regulation. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[5]  D. Gillespie.  The chemical Langevin equation , 2000 .

[6]  S. Mangan,et al.  The coherent feedforward loop serves as a sign-sensitive delay element in transcription networks. , 2003, Journal of molecular biology.

[7]  C Jayaprakash,et al.  The role of dimerization in noise reduction of simple genetic networks. , 2003, Journal of theoretical biology.

[8]  T. Baker,et al.  A specificity-enhancing factor for the ClpXP degradation machine. , 2000, Science.

[9]  C. Pennartz,et al.  A Mathematical Model for the Intracellular Circadian Rhythm Generator , 1999, The Journal of Neuroscience.

[10]  P. Swain,et al.  Stochastic Gene Expression in a Single Cell , 2002, Science.

[11]  P. Swain,et al.  Gene Regulation at the Single-Cell Level , 2005, Science.

[12]  Ertugrul M. Ozbudak,et al.  Regulation of noise in the expression of a single gene , 2002, Nature Genetics.

[13]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[14]  A. van Oudenaarden,et al.  Noise Propagation in Gene Networks , 2005, Science.

[15]  G. Macino,et al.  Role of a white collar‐1–white collar‐2 complex in blue‐light signal transduction , 1999, The EMBO journal.

[16]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[17]  M. Bodnar,et al.  The nonnegativity of solutions of delay differential equations , 2000, Appl. Math. Lett..

[18]  Farren J. Isaacs,et al.  Prediction and measurement of an autoregulatory genetic module , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[19]  J. Miȩkisz,et al.  Stochastic Models of Gene Expression with Delayed Degradation , 2011, Bulletin of mathematical biology.

[20]  A. Arkin,et al.  Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells. , 1998, Genetics.

[21]  Peter Fraser,et al.  Transcription complex stability and chromatin dynamics in vivo , 1995, Nature.

[22]  D. A. Baxter,et al.  Modeling Circadian Oscillations with Interlocking Positive and Negative Feedback Loops , 2001, The Journal of Neuroscience.

[23]  Martin A. Lema,et al.  Delay model of the circadian pacemaker. , 2000, Journal of theoretical biology.

[24]  S. Shen-Orr,et al.  Network motifs in the transcriptional regulation network of Escherichia coli , 2002, Nature Genetics.

[25]  J. Dunlap,et al.  WC‐2 mediates WC‐1–FRQ interaction within the PAS protein‐linked circadian feedback loop of Neurospora , 2001, The EMBO journal.

[26]  K. Cooke,et al.  On zeroes of some transcendental equations , 1986 .

[27]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[28]  Xiaodong Cai,et al.  Exact stochastic simulation of coupled chemical reactions with delays. , 2007, The Journal of chemical physics.

[29]  Steven Henikoff,et al.  Modulation of a Transcription Factor Counteracts Heterochromatic Gene Silencing in Drosophila , 2001, Cell.

[30]  Mads Kærn,et al.  Noise in eukaryotic gene expression , 2003, Nature.

[31]  T. Kepler,et al.  Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations. , 2001, Biophysical journal.

[32]  J. Raser,et al.  Control of Stochasticity in Eukaryotic Gene Expression , 2004, Science.

[33]  Jan Poleszczuk,et al.  Negativity of delayed induced oscillations in a simple linear DDE , 2011, Appl. Math. Lett..

[34]  Josef Hofbauer,et al.  The theory of evolution and dynamical systems , 1988 .

[35]  F. Gantmacher,et al.  Applications of the theory of matrices , 1960 .